2. Leg DesignHexotica - The Design and Implementation of a Small Walking Robot
The successful design of a legged robot depends to a large extent on the leg design chosen. Since all aspects of walking are ultimately governed by the physical limitations of the leg, it is important to select a leg that will allow for a maximum range of motion and that will not impose unnecessary constraints on the walking gait chosen. The first stage of the leg design process therefore consists of a search for an optimal leg design.
2.1 Leg Design Alternatives
A survey of the literature shows that there are a number of different leg designs currently employed for walking robots. All have advantages and disadvantages and a few of the options considered are outlined here. Some of these options are described in more detail in [10] [15] [16] and [17]
2.1.1 Simple two-link leg
Variations on the simple two-link manipulator, shown in Figure 1, were considered as one potential leg design. This design consists of two links connected through a knee joint. The walking motion is accomplished by controlling the angle of the two links to position the end effector, or foot. The entire leg is mounted on a swiveling base in order to advance and retract the leg. There are a number of different ways in which the joints can be actuated, with the actuation of the knee joint being the major difference from one design to the next. Options include mounting the motor at the joint itself or using a pulley, chain or lead screw to set the angle of the knee using an actuator mounted near the base of the leg.
Figure 1 - Simple two-link leg
The major drawback of this design is the necessity to actuate remote joints. Placing the actuator at the knee joint adds various dynamic effects to the leg which have to be compensated for by the controller. This adds complexity to the control algorithms needed to move the leg. It also requires more powerful motors at the hip joint to move the added mass of the leg. Remote actuation, in which the actuators are located at the base of the leg, eliminates some of these problems, at the cost of increasing the complexity of the mechanism. The coupling of the motion of the end effector relative to the actuators is another undesirable characteristic of this leg design.
2.1.2 Mammalian Leg
Another potential leg design considered is modeled after a typical mammalian leg. The basic walking motion is shown in Figure 2. The entire leg swings about the hip joint to advance and retract the leg while the angle of the knee joint is controlled to lift the foot out of the way during the advance phase. The leg can be designed using a four bar linkage structure with one of the links being of variable length.
Figure 2 - Typical mammalian leg stride showing four distinct stages of a step
This design was initially rejected because it is not clear how large a range of motion is available with this leg configuration. The motions are highly coupled and the effective workspace is somewhat limited. Also, the fact that the entire weight of the robot is supported by the hip joint necessitates a large, powerful and potentially expensive motor, which does not conform to the design criteria set out for the leg.
2.1.3 Pentagraph Leg
A pentagraph, or five-bar mechanism, has been used in other robotic applications to provide mechanical transparency of a linkage structure [9] By situating the actuators at a central base and using light members, the dynamic effects of the mechanism can be minimized while still providing structural rigidity. An adaptation of this technology to legged robots, as shown in Figure 3, was considered. The planar five-bar mechanism could be used to position the leg shaft in a plane. The shaft or the entire leg platform could then be controlled to provide motion in the third axis.
Figure 3 - Application of pentagraph to legged robot
One of the major drawbacks of this arrangement is the high degree of coupling of the motion. It is not possible to move the position of the end effector in a particular direction without moving at least two motors simultaneously. This complicates the control algorithm necessary to move the leg.
2.1.4 Pantograph Leg
The pantograph leg design was selected for the initial leg design for the robot. It has proven popular for a number of different legged robots appearing in the literature [10] [16] The pantograph mechanism consists of a simple four bar parallelogram mechanism. This simplifies the kinematics associated with the mechanism and thereby reduces the computational complexity of the control (see Figure 4).
Figure 4 - Leg motion showing effective workspace (dimensions in cm).
The workspace of the leg shows that with this configuration there is still some coupling of the end effector motion, which results in the curved trajectories of the foot evident in the figure. It was initially selected because of the simplified geometry. However, mechanical problems encountered during the construction of the first leg pair forced the leg design to be reconsidered and it was decided that the simple two link leg, as discussed previously, would be adopted as the final leg design.
Actuation of the remote joint was accomplished using a lead screw and the new leg configuration is shown in Figure 5. The measured quantity at the knee joint is the angle ‘q ’ while the controlled element is the length ‘l ’. For small changes, the relationship between these two quantities, as shown in the figure, holds true. Since the controller will be updating its control signal at a high rate, this approximation is sufficient.
Figure 5 - New leg configuration (dimensions in cm).
The CAD drawings for the new leg modules are contained in Appendix G – CAD Drawings. There are still a number of mechanical problems with this leg configuration, the most significant of these being the lead screw actuation of the remote joint and the power of the motor used to lift the body of the robot. The mechanical problems encountered are discussed in section 2.5 Mechanical Problems.
Accurate calculation of the foot position will be necessary when precise foot placement is critical, such as when the robot is navigating in rough or otherwise uncertain terrain. If the robot is given the ability to select its foot placements it must be able to move the leg to the appropriate configuration. The position of the end effector in space (in this case the foot of the robot) can easily be calculated if the angles g , q and b are measured as shown in Figure 6.
Figure 6 - The configuration of the leg
The position of the foot in space is given by Equation 1 (see Appendix A - Kinematic Calculations for the derivation of these results).
Equation 1 - Foot position in XY plane
Based on the kinematic equations of the foot, it is possible to generate the Jacobian matrix of the leg. The Jacobian matrix is created by taking partial derivatives of the position equations with respect to the joint angles. This matrix can be used to relate joint velocities to velocities of the foot. It can also be used to map Cartesian forces acting at the foot into equivalent static joint forces. Equation 2 shows the form of the Jacobian matrix for the pantograph leg and the two leg relationships which can be derived using the matrix.
Equation 2 - Jacobian matrix relations used in the field of robotics [2]
The joint velocities required to achieve a particular end effector velocity can be found by mapping the Cartesian velocity of the foot into joint space using the inverse of the Jacobian matrix. This will be discussed further in the section on low-level control of the foot.
A critical consideration during the design stage is the size of the desired workspace of the leg, and hence the size of the leg. The lengths of the various links determine the height to which the leg can be lifted, the distance that the leg can be extended away from the body and the size of each step.
Since there are no fixed constraints on the size of the robot, other than the fact that a larger robot will tend to incur higher costs, a leg with a maximum step height of approximately 15 cm seems appropriate. This allows the robot to step over small objects and navigate shallow stairs while still allowing for a relatively compact overall size. The link lengths chosen and the workspace of the leg are shown in Figure 7 with regards to the three different motions of the leg.
Figure 7 - Workspace of leg with L1=10cm and L2=15cm.
After selecting the simple two-link leg design, the next task involves the generation of the detailed design in order for the robot to be accurately simulated and constructed. AutoCAD has been chosen as the CAD software of preference due to previous experience with the software and confidence in its three dimensional capabilities. A detailed design will highlight any interference problems or other mechanical complications that may occur before the leg is built. These problems can then be addressed at an early stage.
A key consideration in the design of the leg is centralization of the actuators close to the base of the leg. By situating the actuators about a central point, the dynamic effects of the leg can be minimized. Centralization of the actuators can also reduce the dynamic loads on individual motors since they do not have to move the mass of the other actuators at remote locations.
There are a number of different materials that could be used to construct the robot. Cost is a major factor in the choice of material, but weight plays an even larger role in the overall efficiency of the robot. Aluminum is used as the major structural material because of its light weight and easy machinability. Large areas along the axis of the aluminum leg shafts are milled out to further reduce the weight of the legs. Aluminum members are also used for the construction of the frame. Steel shafts are used in critical portions of the leg such as motor linkages to provide increased strength and durability.
In order for the leg to operate properly, it is important that the joints be well designed. They should provide a low friction joint with minimal backlash. They should also be cost efficient. Bushings are used to meet these design constraints. They were chosen over the main alternative, ball bearings, due to their significantly lower cost and the complexities of mounting ball bearings in a relatively small workspace.
The detailed design for the current model is included in Appendix G – CAD Drawings. Each component has a separate drawing that allows for duplication of individual parts, if necessary. There are a few problems with the current design that will be discussed in the next section. These are documented in order to highlight what must be done in the future to optimize the design.
Currently, the legs cannot lift the weight of the robot. They are able to support the weight of the robot because of a high gear ratio between the motor and the drive shaft. However, the motor that lifts the leg does not have sufficient torque to lift the robot body in the current leg configuration. A more thorough investigation of the motor torque requirements should be conducted through the use of mechanical simulation software (the development of a robot simulation is discussed in section 7.3 Software Simulation). By specifying the exact torque requirements of the motor, an optimal configuration can be realized. Motor characteristics must be obtained in order to determine whether the motors selected are sufficiently powerful for actuation of the leg joints.
A lead screw is used to remotely actuate the knee joint. While this allows the actuator to be mounted near the base of the leg, there are a number of problems with the current configuration. Firstly, the motors selected are not very strong. They can advance and retract the leg but may have difficulty under heavy loading.
The attachment between the motor and the lead screw is also something that needs to be improved. The lead screw is currently attached to the motor shaft via a single set screw. There is no other support for this member at its end. When the leg is extended, this shaft is put into tension and this force is in turn applied axially to the motor shaft, something that may reduce the performance of the motor. A better lead screw attachment should be designed if the lead screw is maintained as the means for actuating the remote joint.
As mentioned earlier, there are a number of different methods by which the remote joint could be actuated. Some consideration should be given to the redesign of this part of the leg and more rigorous design and simulation of the robot prior to building a new leg may improve the final leg design. The pantograph leg configuration considered in the first design iteration may help alleviate this problem if it is properly designed and constructed.
copyright information
Back to home
page. Last updated: April 20th,
1997
